A Cell Model of an Ion-Exchange Membrane. Capillary-Osmosis and Reverse-Osmosis Coefficients

Abstract

The capillary-osmosis and reverse-osmosis coefficients of an ion-exchange membrane have been calculated as the kinetic coefficients of the Onsager matrix within the thermodynamics of nonequilibrium processes and on the basis of the cell model proposed previously by the author for charged porous layers. The membrane is assumed to consist of an ordered set of spherical completely porous charged particles placed into spherical shells filled with a binary electrolyte solution. Boundary value problems have been analytically solved to determine the capillary-osmosis and reverse-osmosis coefficients of such a membrane under the Kuwabara boundary condition imposed on the cell surface. The consideration has been implemented within the framework of small deviations of system parameters from their equilibrium values under the action of external fields. Different particular cases of the obtained exact analytical equations have been studied including the case of a binary symmetric electrolyte and an ideally selective membrane. It has been shown that, for the considered cell model of an ion-exchange membrane, the Onsager reciprocity theorem is violated; i.e., the found kinetic cross coefficients are unequal to each other. The violation is explained by the fact that the reciprocity theorem is valid only for systems implying the linear thermodynamics of irreversible processes, for which generalized flows are equal to zero at nonzero thermodynamic forces.

Theoretical and Experimental Study of Joint Osmotic and Electroosmotic Water Transfer through a Cation-Exchange Membrane

“…In this paper, for the first time, we suggested analytical formulae for determining the osmotic and electroosmotic permeability of a real and ideal cation-exchange membrane as a function of an equilibrium concentration of electrolyte. These formulae were obtained based on the cell model of a charged membrane [ 30 , 31 , 32 , 33 , 34 , 35 ] and methods of nonequilibrium thermodynamics in a linear approximation and have been successfully verified on our own experimental data. The achieved results make it possible to entirely characterize existing and promising types of ion exchange and other types of membranes based on the developed cell model of a charged membrane.…”

“…Kinetic coefficients were calculated in [ 32 , 33 , 34 , 35 ] using the cell model that considers the heterogeneous structure of the membrane. The cell model has been successfully verified on different types of ion-exchange membranes and electrolytes [ 30 , 31 ].…”

“…Formula (6) allows calculating the solvent flux through the membrane when applying both direct current and a concentration drop. At the same time, for the electroosmotic and osmotic permeability of an ideal negatively charged (cation-exchange) membrane, we have the following formulae, respectively, where F 0 is the Faraday constant, are the diffusion coefficients of cations and anions in the electrolyte solution at infinite dilution, m 0 is the macroscopic porosity of the membrane, is the characteristic exchange capacity of the membrane system [ 32 , 33 , 34 , 35 ], k D is the specific hydrodynamic permeability of the ionite grain (gel), µ o is the viscosity of the solution.…”

“…In the general case of non-ideal membranes, formulae that are analogous to (7) and (8) are significantly complicated and can be obtained using expressions (4) and (5) and the kinetic coefficients calculated in [ 32 , 34 , 35 ].…”

“…The experimental data presented in Figure 2 were calculated using the cell membrane model [ 30 , 31 , 32 , 33 , 34 , 35 ] and independently obtained data on the ion-exchange capacity of the MF−4SC p.29 membrane equal to = 1.105 mol/dm 3 and its macroscopic porosity m 0 = 7.7%. The remaining physicochemical parameters of the cell model were found using a special procedure for the one-time minimization of the deviation of the experimental values of the electroosmotic and osmotic transport numbers from their theoretical values.…”